Abstract.
In an earlier article on this website
I surveyed the historical context
of tuning and temperament, concluding with some remarks about the sanctity of
the octave in terms of its tuning purity.This
article continues the story by asking why tempered octaves have seldom been
considered in the long history of tuning keyboard instruments.Although a definite answer is elusive, a probable reason is that
temperaments with impure octaves are difficult to tune by ear, and therefore it
is only recently that the advent both of electronic tuning devices and digital
musical instruments have made them more accessible for study.

Various
temperaments with impure octaves are described, with the octaves tuned both
sharp and flat from pure.The work focuses exclusively on temperaments appropriate for
the organ, because a temperament suitable for this instrument might be less
attractive for others, and vice versa.This is partly because of the sustained nature of organ tones, as well as
the availability of stops at many pitches which other instruments do not
possess.The fact that most stops
constituting an organ chorus are octavely related makes the study of
temperaments with impure octaves uniquely interesting for the instrument.

Three
temperaments are discussed in detail, one using offset octaves and another using
Cordier’s recipe where the octaves are sharpened and the fifths pure.The third temperament is called “Flat Octave 1” and it uses
flattened octaves.This has the
advantage that the significantly sharp thirds in conventional Equal Temperament
and the even sharper ones in Cordier’s temperament can be brought closer into
tune.Some mp3 sound clips are
included.

Some
interesting generalisations are mentioned which appear when using impure
octaves, an important one being that an infinity of equal temperaments become
available instead of there being just one as in the case of pure octave tuning.This fact, that impure octaves enable the exploration of more than one
equal temperament, is exciting both in theory and in practice.It opens a door which has been locked for centuries.All of the temperaments with impure octaves discussed in the article are
equal temperaments, which means they can be used in all keys irrespective of
their different characters.

In
an earlier article on this website [1] I surveyed the
historical context of tuning and temperament, concluding with some remarks about
the sanctity of the interval of an octave in terms of its tuning purity.On the organ the octaves are tuned pure, in contrast to the other
intervals such as thirds and fifths which are invariably candidates for various
degrees of tempering or detuning.Thus
if we follow conventional wisdom, all intervals except the octave can be
tempered.

While
various reasons can be proposed to explain why this cultural paradigm has become
so ingrained since the dawn of human history, at least for the organ, it is by
no means obvious why we still adhere to it so strongly today if we approach the
issue with an open mind.The reason
most often quoted is that the harmonics of two pipes an octave apart do not
coincide neatly in frequency if the octave is not perfectly tuned, and in this
case we will hear the pipes beating.But,
and curiously, those who use this to justify pure octaves never insist that all
the other intervals must also be tuned pure.Indeed they often swing the other way to actually state a preference for
tempered intervals – those which generate beats - to prevent the cloying
sweetness which some say would occur with too many pure intervals within the
octave.This might seem to be
merely making a virtue of necessity, because it is of course impossible to have
all the intervals pure – if it were possible, there would be no temperament
problem in the first place.But the
application of different logic to insist on pure octaves on the one hand yet, on
the other, to accept that any or all the other intervals may be tempered could
be seen as perverse, given the difficulties of devising usable temperaments
which follow as a consequence of having pure octaves.

Nevertheless,
I think there is a good, down to earth, practical reason why the octaves have
been tuned pure for so long, and one meets it forcefully when trying to tune an
instrument by ear to a temperament using impure octaves while retaining a
keyboard whose physical structure repeats every octave.Traditionally, tuning an organ by ear means that one first sets the
desired temperament in the middle octave of a single stop.
Next, the twelve note frequencies so defined are propagated across the
compass of the chosen stop merely by tuning the octaves pure.Finally the remaining stops are tuned pure, note by note, to
the rank on which the bearings have been laid.Using impure octaves would mean that one could not propagate the tuning
of the middle octave across the keyboard, nor could one tune the other stops
pure, just by listening for zero beats between unisons or octaves.

This
would make tuning by ear a long winded and error prone procedure if the octaves
were not pure, to the point where it would verge on the impossible or at least
be deemed impractical for routine use.It
is only in recent decades, with the availability of electronic tuning aids or
with digital instruments, that this situation has changed.Consequently, given that it is now possible to tune keyboard instruments
more easily today than for countless centuries past, this article examines some
implications of allowing the octaves to be tempered, how they should be tempered
and what might be gained.

The
work described here relates specifically to the organ, because it is explained
later why the characteristics of a temperament suitable for this instrument
might be less attractive for others, and vice versa.This is partly because of the sustained nature of organ tones as well as
the availability of stops at many pitches which other keyboard instruments do
not possess.Because most stops constituting an organ chorus are octavely
related, the study of temperaments with impure octaves is uniquely interesting
for the instrument.

I
have not presupposed that you will necessarily like the temperaments described
herein.Many zealous writers on
temperament, including some of the best qualified, appear to see themselves as
Inquisitors by implying that anyone who questions their work is beyond
redemption.Padgham’s description
of the unconverted who exhibit “conservatism, fear of the unknown and
ignorance” [9] is particularly shocking, but
regrettably he is not alone and today temperament remains a subject where
invective often masquerades as scholarship.For my part, I shall simply be content if you find this article
interesting in some way or other.Proselytismis not its intention, as it merely attempts to open some doors which hitherto
have been largely closed.Temperament
is an interesting subject to some, but that does not mean that everyone has to
take the medicine and I fully understand those who do not like it.

Reviewing
the relevant arguments from the earlier article [2],
as a rule students of temperament ignore or dismiss the possibility that the
octaves might play a greater role in their subject than merely marking the
boundaries between successive sets of twelve tempered semitones.Most of them never mention it as an option; they proceed on the basis
that pure octaves are axiomatic and always tune them true.Inevitably, this leads to a subjective tuning rigidity across the compass
of a keyboard instrument regardless of the temperament to which it is tuned.The tempering of the intervals in every octave is the same,
and every note is tuned true with its octave above and below.The results are legion as will now be described.

The
beat frequency of any interval on a keyboard instrument depends on the octave in
which it is played.In other words,
a tempered fifth played in the third octave of the keyboard will beat faster
than if it is played in the second octave, but slower than if it were to be
played in the fourth octave.With any temperament which uses pure octaves, the ratio of
these beat frequencies bears a simple relation to the octaves considered – a
tempered fifth in the third octave beats exactly twice as fast as when it is
played in the second octave, four times as fast as in the first octave, and so
on.These exact and simple beat
frequency ratios also apply to any other interval, no matter how carefully they
might have been mutually adjusted within each octave by adopting a favoured
temperament.

With
a recently tuned organ in which all the octaves are well in tune across the
whole keyboard, and with well tuned octavely-related ranks, this can lead to a
hard, sterile, locked-up type of sound especially when chords are played which
span a significant part of the compass.Not
only are there no beats at all between the octaves, but the beat rates between
similar intervals in different octaves are related by exact integer ratios as we
saw above.The sterility only recedes when the tuning of the instrument
drifts over time, in the course of which the octaves often become slightly
impure.The subjective effect can
be even more noticeable and unpleasant in digital instruments because their
tuning never drifts.Again,
remember that the effects we are discussing are independent of the temperament
actually used; they follow purely because the octaves are locked in frequency.

Such
aural sterility never characterises the orchestra or other ensembles of
instruments because no attempt is made by their players to maintain excessive
purity of intervals to the same extent, and this includes purity of the octaves.

Not
only is the effect of well tuned keyboard instruments potentially more sterile
than that of the orchestra.In addition, the sometimes unattractive subjective experience
due to excessively pure octaves is worse for the organ than for stringed
keyboard instruments.This is
because of the different overtone (partial) structures in the two cases.The overtone frequencies of organ pipes have an exact integer
relationship with each other: the second harmonic of any pipe, sounding the
octave above the fundamental, is at exactly twice the frequency of the
fundamental, the third harmonic (the pure twelfth) is at exactly three times the
fundamental, and so on.Thus all
the harmonics of an organ pipe are locked in phase with each other while it
sounds.Using the terms properly
and rigorously, this is why the overtones in this case must be called harmonics.

The
situation described only pertains when the pipes are sounding in their steady
state speaking regime after the attack transients have died away, which of
course occurs relatively quickly.Therefore if an organ is well in tune so that the octaves are
as exact as possible, the sterility both of the octaves and of the beat ratios
between octaves is amplified subjectively by the lack of numerical freedom in
the frequency ratios among the harmonics of the pipes themselves.The mere fact that the sounds of organ pipes do not die away until the
keys are released adds yet further to the potential subjective hardness of the
overall effect of an organ with well tuned pure octaves.

Stringed
keyboard instruments such as the piano and harpsichord do not usually sound as
sterile, in the sense described, as the organ even when they are tuned as well
as possible.The main reason is
that the harmonics of a struck or plucked string (but not a bowed string) are
actually not harmonics at all; they must be referred to as partials or
overtones, not harmonics, because their frequencies are not exact integer
multiples of each other.As the
sound of a struck or plucked string dies away the overtones, which are slightly
mutually sharp, exhibit beats because of their frequency independence.(This also happens during the attack and release transient phases of
organ pipe speech, but because these phases are of such short relative duration
the effect is dominated by the steady state phase in which the overtones are
true phase-locked harmonics as described above).

Another
disadvantage following from the use of pure octaves is that an opportunity is
missed to ease the tight straitjacket of conventional temperaments.The root problem of temperament is to squeeze that uncomfortable set of
bedfellows called the semitones into an octave in such a way that none of the
intervals between them is grossly out of tune.This is done by making small adjustments to their frequencies (e.g.
making the fifths flat, the thirds sharp, etc).Why not ease this problem a little by making the octaves themselves
adjustable as well?Of course, it
would not be possible to detune the
octaves to such an extent that they themselves became unusable.In this case we would only have replaced one problem by another.Somewhere between the two extremes may lie a solution worth exploring.

Impure
octaves are nothing new in keyboard tuning.Many piano tuners routinely sharpen or "stretch" the octaves
when tuning, though the reasons quoted vary.Some maintain that the beats between the partials during the decay of the
sound are less dissonant (when several notes have been keyed) if the octaves are
tuned slightly sharp towards the top of the keyboard.Others say that it is better to tune the octaves sharp so that they will
in time come better into tune as the string tensions relax slightly.Of the two reasons I incline preferentially to the commonsense nature of
the latter.Other reasons for
stretching the octaves also exist.At
the other extreme though, some tuners tune the octaves pure.So we can learn little from the piano scenario, even ignoring the
physical differences in the way it and the organ produce sound as outlined
above.

Yet
impure octaves would without doubt have occurred from time to time if we accept
the notion suggested in the earlier article [3] in
which some “good” temperaments in the 17th and 18th
centuries might have been discovered partly through serendipity because of the
tuning instability of the old stringed keyboard instruments.This was a consequence of their wood frames which were inconveniently
sensitive to changes in temperature, humidity and ageing [4].But regardless of any interesting results which happenstance might have
thrown up, probably a compelling reason why impure octaves were not embraced by
those who worked on temperament in those days was because of the difficulties of
tuning an instrument in this way – by ear alone of course – and this has
been mentioned already.

The
earlier article mentioned that I planned to investigate the matter in more
detail, specifically for the organ, and the results so far will be described
presently.One issue foreseen at
the outset has been confirmed, in that doing the work with the emphasis on
arithmetic and theory which constitutes current work on temperament would almost
certainly be debarred.This is
because pure octaves underpin the entire concept of temperament as it is
understood today, therefore removing them will also remove the relative
arithmetical simplicity of the subject.If the octaves are no longer pure, the subject could easily
become theoretically anarchic and entirely experiential.Any note on the keyboard could in principle take any
frequency value, and the frequencies actually chosen would then arise solely
through empiricism – trial and error.

To
prevent this unpleasing prospect developing, it was considered desirable to
impose a deterministic rather than an indiscriminate progression of octave
tempering across the keyboard.As an example, a temperament might be set for the lowest
twelve notes, say, then the successive octaves above each one would be tempered
progressively according to certain rules to generate its upper brethren.Or the generating temperament could be set in an octave closer to the
middle of the compass and then propagated in both directions, up and down,
according to the same rules.Each octave could contain a completely different
temperament in principle, though in practice it is probably better to regard the
entire keyboard simply as a collection of notes upon which the notion of a distributed
temperament is to be imposed.

These
ideas, particularly that of a distributed temperament, are perhaps difficult to
accept at first acquaintance.Therefore
it might be appropriate to remind ourselves that temperament is largely a
subjective matter in the last analysis, a matter of what the ear will accept.This is illustrated by the fact that among musicians there exists a
spectrum of attitudes ranging from complete indifference to a neurotic
interest in the subject.Put
simply, this is probably related to what musicians regard as an acceptable
degree of out-of-tuneness, and here again there is no single view.This will now be explored.

As
night follows day, imposing a temperament on a keyboard instrument always means
that some intervals will be better tuned than others.A perfectly tuned interval exhibits no beats between any of the harmonics
of either of the two notes, thus we perceive no wavering, pulsating or throbbing
as long as the notes sound.(In
passing we might observe that the formation and meaning of beats is often
misunderstood, and the phenomena are fully explained elsewhere on this website [8]).

Normally
the octaves are tuned pure in all temperaments.However, within the octave some of the other intervals are out of tune by
an amount depending on which temperament has been used.In Equal Temperament none of the intervals except the octaves are in
tune.For example, all the fifths
are slightly flat and the thirds are considerably sharp, whereas in the
Werckmeister III temperament eight fifths within the octave are perfectly tuned.In turn, the intervals which are out of tune in any temperament govern
which keys are useable and which are unusable.In the 18th century Gottfried Silbermann is said to have used
a temperament in which A flat major is the worst key to an extent which most
would regard as intolerably out of tune, whereas C major is very good [5].In Equal Temperament and some others all keys are useable.

For
musical purposes the amount by which the two notes constituting an interval are
out of tune is usually quantified in three ways.Firstly, if a note deviates significantly from the pitch we expect we can
tell immediately that something is wrong.We
form this judgement on the basis of the absolute frequency of the note, and
those cursed with absolute pitch will tolerate smaller deviations than those
without it.However this situation
is unusual unless the temperament in use is a strongly unequal one with
“wolf” notes which render certain keys unusable, or unless the instrument in
question is badly out of tune anyway.

In
most circumstances we use a second way of deciding whether intervals are
acceptably in tune because our ears form a judgement of the frequencies of the beats when the notes sound simultaneously. In this case the
judgement is based on the relative, rather than the absolute, frequencies of two
notes.Thirdly, but more
laboriously, we can measure or calculate their frequencies and then express the
ratio in units such as cents [6].

However
the last two measurements, involving beats and cents, are not equivalent for
musical purposes, and it is therefore regrettable that many authors proceed as
though they are.The number of
beats per second, the beat frequency, is the same as the difference in cycles
per second between two frequencies. The older frequency unit of cycles per second is
properly written today as Hertz (Hz).Thus a beat of 1 cycle per second between two frequencies means that they
differ by 1 Hz.This is always true
regardless of the musical pitch of the notes which give rise to the beats, that
is, regardless of their absolute frequencies and therefore regardless of where
they lie within the keyboard compass.

On
the other hand, the number of cents between the same two frequencies does
depend on where they lie in the keyboard.The
difference can be appreciated by taking an example.At middle C on a couple of 8 foot organ stops, two nearly in-tune pipes
beating at 1 beat per second are out of tune by 6.6 cents, whereas an octave
above at treble C the same beat rate would mean they are out of tune by only 3.3
cents [7].Extending
the same beat frequency of 1 Hz to the extremes of the organ compass, at bottom
C on a pair of 32 foot stops (such luxury!) the two pipes would be out of tune
by 103 cents, more than a semitone.At
top C on a pair of 2 foot stops the difference would be only a minute 0.2 cents,
one five-hundredth of a semitone.Yet
in all these cases the beat frequency perceived by the ear is the same – 1 Hz.Bear in mind that beat frequencies of 1 Hz or so over the compass would
lead many to conclude that an organ was badly out of tune, and many would insist
on it being significantly reduced.

The
foregoing is admittedly rather laboured.However
it is included to emphasise that when presented with two notes sounding
simultaneously, the ear usually latches onto the beats which might exist – it
does not do esoteric calculations to find the equivalent number of cents.In other words, we make a rapid aural and musical judgement as to whether
intervals are adequately in tune or not on the basis of beat frequencies alone.Therefore when cents are used by modern writers on tuning and temperament
to express frequency ratios between notes, it must always be borne in mind that
the cent values vary dramatically across the keyboard for the same beat
frequency.

This
is not always obvious from the prose of many authors.For instance, Padgham in his book on organ tuning stated that “interval
errors of greater than 10 cents from just values are ... significant” [9].
This is demonstrably untrue to the point of being meaningless for the reasons
just rehearsed.Taking another example, a mathematician with whom I was
corresponding on this topic maintained that “cents are cents – why should it
matter where they are in the keyboard?”.Clearly, specialists who allow their lives to revolve too closely around
cents can lose sight of the point that the ear perceives only a beat frequency,
not the difference in cents, when an interval is slightly out of tune.There is no unique correspondence between the beat frequency and its
value in cents, because the latter depends on the position of the interval
within the keyboard compass of the instrument in question.When we leave the comfortable confines of pure octaves and move into the
realm of distributed temperaments across an entire keyboard, the issue assumes
particular importance.

Even
ignoring distributed temperaments, the discussion above is no mere academic
nicety.One important practical
consequence arises in connection with today’s synthesiser-based musical
instruments such as digital pianos, harpsichords and organs.Not all of them can be tuned once purchased, but of those which can the
tuning precision away from Equal Temperament is commonly limited to 1 cent.The fractions of a cent necessary for accurate tuning are unavailable in
these instruments. This means that at the top of a 2 foot stop the notes cannot
be tuned better than about 5 beats per second either sharp or flat from their
Equally Tempered pitch.This is at
least ten times worse than the accuracy a pipe organ tuner would aim for.Thus such instruments can sound rather rough and coarse when high pitched
mixtures and mutations are in use, particularly when certain unequal
temperaments have been set.

Because
many workers on temperament today use digital instruments of one sort or another
to assess their results, whether they admit it or not, this restriction is of
more than passing interest and it is something I have been forced to keep
constantly in mind during the work now to be described. And to forestall the
obvious question, I do use digital keyboard instruments myself.They are very useful for temperament research, but only if one works
consciously within their limitations.

The
same remarks apply to those tuning meters or ETD’s (electronic tuning devices)
where the precision available is limited to one cent.I cannot see that such items are other than a waste of money for tuning
the higher notes of any instrument.

Probably
the simplest way to imagine tuning an organ with impure octaves is merely to
impose a frequency offset on the notes of each octave.Thus the twelve notes in each octave are shifted slightly in frequency en
bloc with respect to their neighbours in adjacent octaves.Because of its conceptual simplicity this method will be described first.

There
are several ways to achieve offset octave tuning, one of which is to tune the 8
foot middle octave (middle C and the 11 notes above it) to the desired
temperament in the usual way and then impose the offsets while tuning the
octaves above and below.This
method has the advantage that the A in the middle octave can still be tuned to
whichever pitch standard is desired (A = 440 Hz or whatever).

The
frequencies of the remaining octaves are then adjusted so that slow beats arise
between corresponding notes in adjacent octaves.For example, treble C would be adjusted so that a slow beat (say around
one beat in three or four seconds) was allowed to remain when sounded with
middle C.It does not matter
whether treble C is adjusted to be slightly flat or slightly sharp, nor does the
exact beat frequency matter either.The
C above treble C would next be adjusted similarly by sounding these two notes
together.In this case it is probably best to tune this octave sharp if
the treble C octave was flat relative to middle C, and vice versa.This will prevent a “runaway” condition in which the extremes of the
keyboard become excessively out of tune compared to the pure octaves case. This
could happen if all the octaves were sharpened, or all of them were flattened,
relative to their neighbours.Alternating
the sharpening or flattening every other octave will prevent this.The C’s below middle C would be treated similarly, offsetting them
alternately flat and sharp as before to achieve a similar beat rate across the
compass.

Within
each octave the notes are tuned using the same temperament for each.Provided the temperament was set accurately in the middle octave, it will
usually be quickest simply to tune corresponding notes in adjacent octaves one
by one such that the beat frequencies imposed on the C’s apply approximately
to the other notes as well.

The
question then arises how to treat stops of other pitches, because the foregoing
related only to an 8 foot unison stop.A
way to proceed is to consider an extension organ, and in this case the answer is
simple -the impure octaves will
appear automatically for all the derived pitches once they have been set across
an extended rank.Therefore, for reasons of compatibility in a 'straight'
(non-extended) organ, it is logical to tune each stop separately against the unison rank,
setting the octave pitches slightly impure on a note by note basis as before.

Fifth-sounding
ranks (such as nazards and twelfths) and third-sounding ranks (such as tierces
and seventeenths) in mixtures and mutations can be tuned true to unison pitch as
in normal practice.This means
tuning them so there is no beat between the third harmonic of the unison and the
twelfth, or between the fifth harmonic of the unison and a tierce.

The
difficulty encountered in tuning an organ by this means will depend largely on
the experience of the tuner.Reduced to its simplest form, it merely means that instead of
tuning octavely-related pipes pure, they are all tuned so that a slow beat
remains.It would be a relatively
straightforward, if time consuming, matter to convert an organ normally tuned
with pure octaves into one with offset octaves at its next tuning if one wished
to assess the effects for oneself.If
one did not like it, it could be converted back again.

In
my first foray into the realm of impure octaves I applied this method of tuning
some years ago to the electronic organ pictured
on the home page of this site. My
"Dorset Temperament" is also used in this instrument, a modest
perturbation of Equal Temperament which introduces a hint of key colour while
still allowing all keys to be used [10].It is no doubt meaningless to describe the results in words, although
adjectives such as “mellow” and “warm” spring to mind.However some sound clips recorded on this instrument are usually to be
found on the home page which might assist those interested to perceive a flavour
of the outcome.The fact I have
been able to live with this frequently-played instrument for some years is an
indication that it is, at worst, not positively objectionable.

The
offset octave tuning just described was an interesting exercise and it provided
useful experience when taking a first step into the subject of impure octaves.However it was an end in itself – that was it.No general insights into the subject could be derived from it and in that
sense it was a blind alley, an intellectual cul-de-sac.

However,
if we recall that Equal Temperament is characterised by twelve equal semitone
steps to the pure octave, equal in the sense that the frequency ratios of
adjacent notes are the same, we can immediately get much further by realising
that we can have equal semitones of any size we choose if the octave is not
pure.In that case they do not need
to be restricted to the frequency ratios of Equal Temperament.This insight is powerful because it enables us to develop an indefinite
number of new temperaments using impure octaves in the manner to be described.

In
Equal Temperament with its pure octaves, the frequency ratio for adjacent
semitones anywhere within the keyboard is 21/12
or the 12th root of 2.As an
example, the frequency of middle C at 8 foot pitch is 261.63 Hz, and that of the
C# above it is 277.18 Hz.The ratio
of these numbers is approximately 1.06, the same as 21/12.(Although
the frequencies used here correspond to the usual pitch standard of A =440 Hz, the same result will be obtained for any other standard).

The
number “2” which appears in the phrase “12th root of 2” simply
represents the fact we traditionally use pure octaves whose frequencies are
related by a factor of exactly 2.If the
octaves are impure then their frequencies can be related by any factor we like.But in that case how can we proceed to derive the corresponding semitone
frequencies?The springboard to
progress in this case is provided by the physical keyboard itself, that
venerable collection of black and white levers of which there are 12 to the
octave.The number “12” which appears in “12th root
of 2” simply represents the fact we have 12 keys or semitones to the octave.It would be a brave mortal who suggested that we use a different number,
stand fast the fact that some workers in temperament have dared to do so, and I
certainly do not have their courage.Therefore
we shall continue to assume a standard keyboard with 12 notes to the octave for
the remainder of this article.

In
this case, retaining 12 notes to the octave but with the octave now defined by
any frequency ratio we like, the frequency ratio for the two notes constituting
a semitone interval anywhere in the keyboard becomes x1/12
or the 12th root of x, where x is the frequency ratio of the
impure octave.Because x can
take any value in principle, we now have potentially an infinite number of
temperaments in each of which the semitone frequency ratios are equal, and thus
an infinite number of equal temperaments with impure octaves.This contrasts with the case for x = 2 (pure octaves)
where there is only a single Equal Temperament.Infinity is an over-used word and in the present situation it
has little meaning in practice, because most of the possible equal temperaments
will be unusable because of gross dissonances.Qualitatively this is no different to the conventional case when trying
to derive usable temperaments using pure octaves.Nevertheless, the fact that impure octaves allow us to
explore more than one equal temperament is exciting both in theory and in
practice.It opens a door which has
been locked for centuries.

Note
the deliberate use of a particular nomenclature in the foregoing.In this article Equal Temperament, spelt with upper case
‘E’ and ‘T’, means the one and only Equal Temperament which is possible
if the octaves are pure.Using
lower case letters, ‘e’ and ‘t’, indicates one of the many equal
temperaments which are possible once the octaves become impure.The distinction is not mere pedantry as it is easy to get confused if we
do not keep the differences in mind.

It
is also easy to get confused if cents are used when describing temperaments with
impure octaves.So beware of cents
yet again!The confusion can arise
if we do not remember that the cent is a measurement derived from Equal
Temperament using pure octaves.Unfortunately
it is often necessary to use it when tuning an instrument with impure octaves
because electronic tuning meters often give readings in cents away from Equal
Temperament.Similarly, most if not
all digital musical instruments also require temperaments to be set in terms of
cents.Whether we like it or not,
conventional Equal Temperament has become the de facto standard to which
all others are referred when tuning a keyboard instrument and therefore it would
not be helpful to redefine the cent in what follows.

A
temperament with pure fifths is an obvious idea which springs to mind when we
recall one of the main problems of conventional temperaments with pure octaves.Keeping the octaves pure means that some or all of the fifths have to be
flattened so they can fit into the octave.If we stretch the octaves to make them slightly sharp, this can be
avoided and all the fifths can then be made pure.The idea is obvious if only because it has been cast in the guise of a
fundamental problem by virtually every writer on temperament for centuries past,
and the diagrams and explanations in reference [9] are
as good as any.It is therefore
less obvious, indeed it is surprising, why some of the same writers have not seen that
the problem could become a virtue by developing temperaments based on pure
fifths rather than on pure octaves.

Thus
instead of the octaves being pure with twelve semitones tempered in one way or
another, the fifths are now pure with seven, slightly larger, semitones.Twelve of these larger semitones make up the new, larger, octave.If we choose all the semitones to have the same frequency
ratio one with the next, this temperament becomes one of the class of equal
temperaments using impure octaves referred to above.

Numerical
data for some intervals in conventional Equal Temperament (with pure octaves)
compared with a temperament constructed in this way (with pure fifths) are in
the table below.The numbers are
given to 6 figure precision to minimise rounding errors if you use them in your
own calculations.

Parameter

Pure
Octaves

Pure
Fifths

Octave:
frequency ratio (freqs in Hz) (i.e. ‘ x ’ in x 1/12)

2.00000

2.00388

Octave:
frequency ratio (cents)

1200.00

1203.35

Beat
frequency between A440 & A above (Hz)

0.00000

1.70521

Fifth:
frequency ratio (freqs in Hz)

1.49831

1.50000

Fifth:
frequency ratio (cents)

700.000

701.955

Beat
frequency between A440 & E above (Hz)

1.48984

0.0000

Fourth:
frequency ratio (freqs in Hz)

1.33484

1.33592

Fourth:
frequency ratio (cents)

500.000

501.397

Beat
frequency between A440 & D above (Hz)

1.98861

3.41042

Major
third: frequency ratio (freqs in Hz)

1.25992

1.26073

Major
third: frequency ratio (cents)

400.000

401.117

Beat
frequency between A440 & C# above (Hz)

17.4611

18.8924

Semitone:
frequency ratio (freqs in Hz) (i.e. x 1/12 )

1.05946

1.05963

Semitone:
frequency ratio (cents)

100.000

100.279

Table
1.Some data for intervals in two
equal temperaments using pure octaves and pure fifths.

The
table shows that, using the nomenclature of the previous section, x in
the expression x1/12now
takes the value 2.00388 instead of exactly 2 for conventional Equal Temperament.The factor x is the frequency ratio of notes separated by an
octave, and in Byzantine units (which I detest) it means each octave has been
sharpened by one seventh of the Pythagorean Comma.Recall that the cent still retains its usual meaning of one hundredth of
a semitone in conventional Equal Temperament with pure octaves.It does not mean one hundredth of a semitone in this alternative
temperament with pure fifths.

The
beat frequencies in the table relate to the most obvious (and slowest) beats
which the ear would usually perceive between two principal-toned stops sounding
the intervals specified.In the
case of the octave, this beat arises between the fundamental of the upper note
and the second harmonic of the lower.For
the fifths it arises between the second harmonic of the upper note and the third
harmonic of the lower.For the
fourths it arises between the third harmonic of the upper note and the fourth
harmonic of the lower.For the
major thirds it arises between the fourth harmonic of the upper note and the
fifth harmonic of the lower.

A
few authors have proposed temperaments using pure fifths from time to time, and
that due to Serge Cordier is probably the best known [11].I am grateful to a French speaking correspondent [12]
for pointing out the happy pun on his name, whose meaning can be bent to denote
one who makes or mends strings.In
view of this it is apposite that Cordier’s work relates primarily to the
piano, though my correspondent suggested it may also have been tried on the
organ [13].On
the piano there seems little doubt that a temperament with pure fifths has merit
in the opinion of some whose judgement most of us will respect if the following
quotations can be taken at face value [14].Speaking of his own Steinway which had been tuned according to
Cordier’s scheme, Lord Yehudi Menuhin was reportedly impressed after a concert
during which he played the violin while his sister Hephzibar accompanied him
on this piano.Apparently he said
“I have never heard a piano sound so free, with such a rich tone”.And Paul Badura-Skoda apparently said “All that is great is simple.This fundamental truth applies also to the new system of tuning of Serge
Cordier, who obtains astonishing results by means of just fifths and
imperceptibly stretched octaves ”.

Several
variants on Cordier’s scheme apparently exist and for these the literature
should be consulted.This article
describes the application of a pure-fifths temperament to the organ rather than
the piano or any other instrument.

Putting
the arithmetic aside for a while, some of the more obvious aural and musical
effects of a temperament with pure fifths will now be described.The temperament is defined by the parameters in Table 1 above and its
most important musical attributes are:

1.
The fifths are all pure.At first
sight this might be thought to have beneficial implications for the quint ranks
in mixtures and mutations.However
while the fifths within an octave are indeed pure, extended quint intervals such
as twelfths which cross octave boundaries are not.Therefore one will not obtain quite the effects from mixtures with this
temperament which one gets (in certain keys) with the same mixtures in
pure-octave unequal temperaments which also contain some pure fifths.But nor will the mixtures sound quite the same as they do in conventional
Equal Temperament with no pure fifths.

2.
The fourths are all sharp by the same amount (3.35 cents), but the beats vary in
frequency depending where they are within the keyboard compass.Table 1 shows that the beat frequency between A 440 and the fourth above
is a rather uncomfortable 3.4 Hz at 8 foot pitch, nearly twice as fast as in Equal Temperament.
This is a disadvantage of this temperament.

3.
The octaves are all slightly sharp by the same amount (3.35 cents), but the
beats between them vary in frequency depending where they are within the
keyboard compass.Table 1 shows
that the beat frequency between A 440 and its octave above is about 1.7 Hz at 8
foot pitch.

4.
The major thirds are slightly sharper than in conventional Equal Temperament,
the difference between them in the two temperaments being 1.12 cents.The difference is negligible in practice considering that the thirds are
significantly out of tune in Equal Temperament in any case, as shown by Table 1.However this does not mean that the sharpened thirds should be regarded
as any less objectionable than in Equal Temperament, and the sometimes
unsatisfactory effect of third-sounding mutations (tierces etc) in Equal
Temperamentwill persist with this
temperament.

5.
All keys can be used.

6.
In theory there is no key colour or key flavour between the different keys
because we are using an equal temperament in which all adjacent semitone steps
are equal in terms of their frequency ratio.This attribute is the same as in conventional Equal Temperament.

7.The organ exposes the impure tuning between various stops at several
octavely-related pitches in a unique manner not available with the piano or any
other instrument (except maybe a large harpsichord also having stops at
different pitches).The temperament
is therefore well suited to the organ in this respect.

With
impure octaves on the organ the question arises how to treat stops of other
pitches, a problem which does not arise with the piano.The same approach is adopted here as for the offset octaves case
considered previously, and it will now be repeated for convenience.

In
conventional temperaments with pure octaves, octave pitches at 4 foot, 2 foot,
etc are simply tuned true with the 8 foot ranks.However in this case an answer to the tuning question presents itself if
we consider what would happen with an extension organ, in which an extended rank
had been tuned with impure octaves.The
impure octaves would then appear automatically for all the pitches derived from
this rank.For reasons of
compatibility when extension is not used, as in a ‘straight’ organ, it
therefore seems desirable to tune the various ranks in the same way.Therefore it will be necessary to tune each stop separately against the 8
foot rank, setting the octave pitches slightly impure on a note by note basis in
the same way they would appear if derived from a single extended rank.

Fifth-sounding
ranks (such as nazards and twelfths) and third-sounding ranks (such as tierces
and seventeenths) in mixtures and mutations can be tuned true to unison pitch as
in normal practice.This means
tuning them so there is no beat between the third harmonic of the unison and the
twelfth, or between the fifth harmonic of the unison and a tierce.

It
is not straightforward to set up this temperament by ear.At first sight it might be assumed that one would set the pitch standard
of the appropriate note (e.g. A = 440 Hz), and then simply tune a principal-tone
stop from it by walking up and down the keyboard using pure fifths alone.In fact the five octave compass of the organ keyboard means that one
could not reach the necessary twelve fifths, for which one would need at least
seven octaves.Of course, the
fifths and fourths in the middle octave could be tuned in the usual way, tuning
the fifths pure but tempering the fourths 3.35 cents sharp, and in fact this
would be easier and quicker than tuning them to Equal Temperament in which all
twelve intervals would need to be tempered.However, propagating the temperament across the range of the keyboard by
tuning corresponding notes in the various octaves would then require all of them
to be tempered note by note because the octaves are not pure.The whole process would then need to be repeated for stops of other
footages on a stop by stop and note by note basis because these could not be
tuned pure to the datum rank.

An
experienced tuner might not be daunted by these problems, but others would
probably find it easier to use an electronic tuning meter.To assist this process a Microsoft Excel tuning chart can be downloaded here
of which an extract is shown below:

The
chart has five sets of columns, each set structured as in Table 2 above.This illustrates tuning data for the bottom octave of an 8 foot rank, and
the complete spreadsheet continues the data downwards to cover five octaves.Together with the other four sets of data, pitches from 32 foot to 2 foot
are covered for a five octave compass.

The
user can type the desired pitch standard into the spreadsheet (e.g. A = 440 Hz),
whereupon the data adjusts itself as necessary.For each note at each pitch, Table 2 shows that the absolute note
frequencies are tabulated together with their deviations in Hz and in cents from
conventional Equal Temperament.This
should provide sufficient data to be compatible with most tuning meters and the
frequency synthesisers in digital organs.

In
the final analysis the aural and musical effect of any temperament is what
matters.By far the best way to
assess this temperament is to set it up on an organ, either piped or digital,
and play on it and with it over a considerable period.Nothing which can be said or described here can approach its feel and
capabilities, its advantages and drawbacks.However a couple of mp3 excerpts are appended which might give some
slight flavour of how it sounds.

The
piece in each case is an excerpt from J S Bach’s Prelude in E flat (BWV 852)
from book 1 of his Well-Tempered Clavier collection, played firstly on
conventional Equal Temperament with pure octaves and then on this temperament
with pure fifths. This piece was chosen because it is an interplay between
melodic and chordal elements from the outset and therefore it enables the
temperament to be heard in both kinds of music. The excerpts were played on a
small digital chamber organ with the following stop list, tuned to the two
temperaments in turn:

Gedackt

8

Rohrflöte

4

Principal

2

Quint

1
1/3

Octave

1

The
8, 4, 2 and 1 foot stops were used together so that some idea can be gained of
how the temperament sounds in a chorus with multiple octavely related pitches,
which of course are not tuned pure one with the other in this temperament.

(played
successively in two equal temperaments, first using pure octaves then pure fifths)

Some
subjective remarks about the temperament include the fact that it sounds
significantly warmer, richer and better in tune than I had anticipated before
actually trying it.The organ
throws up the attributes of this temperament differently, indeed probably
better, than does the piano because the various stops at several octave pitches
– which the piano does not have - are not tuned pure as they would normally be
as discussed already.The
temperament is therefore well suited to the organ in this respect at least.It also has some of the pleasing characteristics one experiences with
certain unequal but “good” temperaments with pure octaves.For instance, from time to time it results in some unexpected but
attractive mild dissonances, though these appear uniformly in all keys because
this is an equal temperament in that sense.In summary, perhaps the best way to regard this temperament is to imagine
a version of conventional Equal Temperament with less sterility and more warmth
than the ordinary one.

There
is another factor unique to the organ which has relevance when assessing how any
temperament sounds, not just this one.Because
virtually every organ has a different stop list, and therefore a different
selection of registers having different timbres and pitches, one has to be
cautious when accepting or rejecting a temperament for the instrument.Unlike those other instruments which are often tuned to different
temperaments, such as the harpsichord or the clavichord, one cannot generalise
so easily with the organ.Thus,
what sounds well on one organ might be less attractive on another, or what is
good for the clavichord or harpsichord might be intolerable for the organ and vice
versa.This is because the
different stops on a particular organ have different numbers and strengths of
harmonics, and the huge number of beats which arise between all the harmonics of
all the stops of all the notes in use at a given instant are largely responsible
for the subjective effect of a given temperament.This effect will be unique for each and every organ, and even for a
single instrument it will vary depending on the stops being used.

I
have demonstrated many times the remarks just made, at least to my own
satisfaction.With the example
above, the effect of this temperament on this particular digital organ varies
depending on which stops are drawn.If
the 1 foot stop is put in and the same piece then played again on the 8, 4 and 2
foot stops only, the difference between Equal Temperament and this one with
impure octaves is less marked.It
still sounds warmer and richer, but the relatively fast beats which arose
between the 1 foot stop and the others are now absent.

We
have already seen that there is potentially an indefinite number of equal
temperaments using impure octaves, depending on the value selected for the
parameter x in the quantity x1/12wherex is the frequency ratio of two notes an octave apart.If the octaves are pure x takes a value of exactly 2, and if they
are sharp (as in Cordier's temperament described above) it is greater than 2.In
principle there is no reason why one should not flatten the octaves by using x
with values less than 2.

x1/12is the
frequency ratio of the twelve semitones within an octave in the case where they
are equal.Again, there is no
reason in principle why one has to temper them equally, and by choosing
frequency values empirically for the semitones (as is done in conventional
unequal temperaments with pure octaves) yet further opportunities arise for an
additional range of unequal temperaments with impure octaves.

Therefore
some remarks follow about other temperaments which can be developed on this
basis. The discussion is brief, partly to keep the length of this already long
article within reasonable bounds, and partly because work in these areas has not yet
progressed far enough to report it more fully.

The
temperament discussed above used sharpened or stretched octaves, and it rather
goes against intuition to consider the opposite situation in which the octaves
might be flattened.In such cases
the value of x will be less than 2, and the frequency ratio of adjacent
semitones will be x1/12for equal tempering.The
counter-intuitive aspect is because the history of temperament is dominated by
the battle to squeeze the intervals into a pure octave in a way that minimises
the tuning problems caused by flattened fifths, and making the octaves even
smaller might be thought to exacerbate the problem.Nevertheless, the flexibility of impure octaves means that one can
investigate temperaments based on flattened octaves as well as sharpened ones to
discover what the ear thinks of them.

A
potential advantage of flattening the octave is that the thirds, considerably
sharp in conventional Equal Temperament and even sharper in temperaments such as
Cordier’s, can be brought better into tune.It is not possible to bring them exactly into tune while retaining equal
tempering of the semitones because then the fifths and octaves will become
unacceptably flat.However one
approach is to choose a compromise tuning in which the frequencies of the
thirds, fifths and octaves are all out of tune by about the same amount.

Considerable
juggling with the numbers is possible when searching for a useable temperament
with flattened octaves, but an interesting convergence appears if the octave is
flattened by 11.16 cents.In this
case the fifths will be 8.47 cents flat and the thirds 9.97 cents sharp.Thus the tempering of all three intervals is now approximately the same,
measured in cents.In round figures
they are all tempered about 10 cents away from pure.This temperament will be called “Flat Octave 1” for
convenience.For comparison
purposes, Table 3 below shows the tuning of these intervals in two other
temperaments as well – conventional Equal Temperament (pure octaves) and the
Cordier temperament discussed earlier (sharpened octaves):

Interval

Pure
octave:

Equal
Temperament

Sharp
octave:

Cordier

Flat
octave:

“Flat
Octave 1”

Octaves: cents from pure

0.00

+3.35

-11.16

Fifths: cents from pure

-1.96

0.00

-8.47

Major thirds: cents from pure

+13.69

+14.81

+9.97

Size of semitone (cents)

100.00

100.28

99.07

Table
3.Comparison of some intervals in
various equal temperaments (cents)

In
Table 3 the sizes of the 3rds, 5ths and octaves are shown by the amount they
deviate in cents from pure.However,
as described at length earlier in this article, what matters more to the ear are
the beat frequencies of these intervals, and Table 4 tabulates these for each
one together with the interval of a fourth:

Interval

(lower
note A440 Hz)

Pure
octave:

Equal
Temperament

Sharp
octave:

Cordier

Flat
octave:

“Flat
Octave 1”

Octave: beat freq A440 - A (Hz)

0.00

1.71

5.65

Fifth: beat freq A440 - E (Hz)

1.48

0.00

6.44

Fourth: beat freq A440 – D (Hz)

1.99

3.41

2.74

Major third: beat freq A440 – C# (Hz)

17.48

18.89

12.70

Table
4.Comparison of some intervals in
various equal temperaments (beat frequencies in Hz, with the lower note of each
interval being A = 440 Hz)

The
beat frequencies vary depending where the intervals lie in the keyboard, and the
values in Table 4 are for the middle octave of an 8 foot stop with the lower
note of each interval being A = 440 Hz.We
see for the Flat Octave 1 temperament that, although the 3rds, 5ths and octaves
are detuned by approximately the same number of cents (Table 3), the beat
frequencies are not the same (Table 4).Nevertheless
the differences between the beat frequencies for these three intervals have been
evened out to some extent for Flat Octave 1 compared to the other two
temperaments.In particular, the
fast beat of the thirds in Equal Temperament and Cordier’s temperament has
been considerably reduced in Flat Octave 1.This is potentially important because it is the significantly sharpened
thirds which make conventional Equal Temperament sound rather coarse to many
ears, and Cordier’s temperament is no better (indeed slightly worse) in this
regard.

In
Flat Octave 1 none of the beats between the 3rds, 5ths and octaves are anything
like as fast as the fastest in the other two temperaments.Moreover, and importantly, the fourths are much better in tune than the
thirds, fifths and octaves, having a beat frequency between A 440 and D of only
2.74 Hz.They are therefore much
better in tune than in Cordier’s temperament and they beat only slightly
faster than in Equal Temperament.

The
important question is – what does the ear make of Flat Octave 1?As with all temperaments, no amount of theory and arithmetic can answer
this and it can only be judged by tuning an organ to it and trying it.Having done this with a digital instrument, the main subjective features
include the following:

1.Playing on a single 8 foot principal-toned stop, the temperament sounds
attractive on the whole - to me.It has
what might be described as a somewhat quaint, olde-worlde flavour without being
too out of tune over most of the keyboard (but see 4 below).The subjective difference between it and Equal Temperament (pure octaves)
is more marked than for the Cordier temperament (sharpened octaves) discussed
earlier.

2.The thirds are noticeably smoother than in Equal Temperament and
Cordier’s temperament.

3.The fourths are much better in tune than the thirds, fifths and octaves.They are also much better in tune than in Cordier’s temperament.

4.Towards the top of the keyboard on an 8 foot stop, and with higher
pitched stops lower in the keyboard, the beats between the significantly
flattened fifths and octaves become excessively prominent in my opinion. This
is, of course, because their beat frequencies increase with the pitch of the
notes.A solution to this problem
is to progressively reduce the amount of flattening of the higher octaves as the
pitches of the notes increase.This
will produce a genuine distributed temperament, one whose characteristics vary
across the compass from octave to octave.

5.This is an equal temperament and thus all keys can be used, but for the
same reason there is no key colour.

I
have found the Flat Octave 1 temperament is sufficiently attractive to
encourage me to play on it at length, comparing it subjectively with the other
temperaments described in this article as well as more conventional ones.The excessively sharp thirds and fourths created by sharpening the
octave, as in Cordier’s temperament, are less objectionable with this one
where they are better in tune.A
disadvantage is the noticeable beats of the fifths and octaves at higher
pitches, though this could be improved by reducing the amount of octave
flattening towards the top of the compass as mentioned in 4 above.On balance this temperament therefore merits further study in my view.So watch this
space!

Unequal
temperaments are those in which the ratio of the frequencies of adjacent
semitones can take any value from note to note, unlike equal temperaments in
which the ratio remains constant.These
definitions of “equal” and “unequal” hold for all temperaments,
regardless of whether the octaves are tuned pure or impure.

With
pure octaves, unequal temperaments are often lauded in comparison to the one and
only Equal Temperament which is possible in that situation.In a sense this is only to be expected because there is theoretically an
infinity of unequal temperaments, and it is therefore unsurprising that there is
a correspondingly inexhaustible treasure trove of experience to be discovered.One factor characterising the unequal temperaments which does not apply
to Equal Temperament is that they possess key colour, a subjective flavour which
heightens the effect of modulation into different keys.In fact it is an excess of key colour which makes certain keys unusable
in some unequal temperaments because of the presence of “wolf” intervals
which are grossly out of tune.

The
same applies to equal and unequal temperaments which use impure octaves in that
the equal temperaments, such as those described in this article, do not have key
colour whereas the unequal ones will have.The major difference between the pure and impure octaves situation, as we
have observed already, is that there is an indefinite number of both unequal and
equal temperaments, unlike in the case of pure octaves in which there is only
one Equal Temperament.

Therefore
the obvious next step is to investigate unequal temperaments using impure
octaves if only because these will possess key colour which the equal
temperaments do not.Currently I
have not gone this far.However the
availability of impure octaves ought to give an additional degree of freedom
when developing unequal temperaments which does not apply if the octaves are
pure, consequently the unequal temperaments which can be developed would likely
be original in the sense they had not been heard before.

There
is an extra subjective dimension to the problem of devising unequal temperaments
which does not apply to equal ones, and this relates again to the key colour
issue.One cannot proceed until one
has decided on the desired intonation of the 24 keys (12 major and 12 minor) in
terms of the purities or otherwise of the intervals they contain.For instance, one might decide that certain keys must contain thirds as
nearly pure as possible.Such
constraints as these in the unequal temperaments using pure octaves with which
we are familiar enable us to select which unequal temperament to use.However with impure octaves the problem is posed the other way round –
instead of selecting a temperament which already exists because its intonation
is pleasing in some way, we have to develop the temperament after having first
decided which intervals and in which keys give the pleasurable effects which are
sought.

Until
we answer this question to our satisfaction we cannot proceed, but the answer
will doubtless be different for different individuals.In this sense there cannot be a single unequal temperament using impure
octaves which will please everybody, just as with the pure octaves case.

The
work described in this article began by asking why tempered octaves have seldom
been considered in the long history of tuning keyboard instruments.Although a definite answer is elusive, a probable reason is that
temperaments with impure octaves are difficult to tune by ear, and therefore it
is only recently that the advent of electronic tuning devices has made them more
accessible for study.In the same
timescale the appearance of digital musical instruments has facilitated
experiments with these and other temperaments.It is therefore perhaps not coincidence that one of the few existing
temperaments using impure octaves, that due to Cordier, did not arise until the
1970’s.

This
article has described research on temperaments with impure octaves, tuned both
sharp and flat.The work focused
exclusively on temperaments appropriate for the organ, and it was pointed out
that the characteristics of a temperament suitable for this instrument might be
less attractive for others, and vice versa.This is partly because of the sustained nature of organ tones as well as
the availability of stops at many pitches which other instruments do not
possess.The fact that most of the
stops constituting an organ chorus are octavely related makes the study of
temperaments with impure octaves uniquely interesting for the instrument.

Three
temperaments using impure octaves were described, one using offset octave
tuning, and a second using Cordier’s recipe where the octaves are sharpened and
the fifths pure.Disadvantages of
the latter include the thirds which are even sharper than in Equal Temperament,
and the fourths which are much sharper.The
third temperament used flattened octaves.Called
Flat Octave 1, this had the advantage that the significantly sharp thirds in
conventional Equal Temperament and the even sharper ones in Cordier’s
temperament could be brought closer into tune.The fourths were also much better in tune than in Cordier’s
temperament.A disadvantage of Flat Octave 1 is the detuned fifths and
octaves which become noticeable because of their beat frequencies at higher
pitches.This could be overcome by
progressively reducing the amount of octave flattening towards the top of the
keyboard, thereby producing a distributed temperament whose characteristics vary
across the compass.

Some
interesting generalisations were mentioned which appear when using impure
octaves, the most important being that an infinity of equal temperaments become
available instead of there being just one in the case of pure octave tuning.All of the temperaments with impure octaves discussed in the article are
equal temperaments, which means they can be used in all keys irrespective of
their different characters.

The
next step is to investigate unequal temperaments using impure octaves if only
because these will possess key colour which the equal temperaments do not.The availability of impure octaves ought to give an additional degree of
freedom when developing unequal temperaments which does not apply if the octaves
are pure, consequently the unequal temperaments which can be developed would
likely be original in the sense they had never been heard before.

4.One correspondent insisted that his harpsichord and clavichord exhibited
negligible tuning instability over many months and that serendipity could
therefore have played no part in the development of Baroque temperaments as I
had suggested.However he failed to
remember that his instruments reside in a centrally heated and air conditioned
room, which would have reduced environmental effects on their wooden frames to
insignificance once they had acclimatised to the conditions.Werckmeister, Bach and their contemporaries had no such luxuries to
offset the climatic extremes where they lived in continental Europe.

5.
This has led some to state that J S Bach avoided writing his major organ
works in A flat major so they could be played on Silbermann’s organs.One only has to examine the way his harmonies modulate through various
keys to see the fallacy of this argument.For
instance, A flat major is an important element of the harmonic texture in the
‘great’ Prelude in C major (BWV 547a).On Silbermann’s temperament this piece, while perhaps not unplayable,
sounds gross in places to most ears.

6.In the same way that an Equally Tempered semitone represents a twelfth
part of a pure octave, a cent represents one hundredth of an Equally Tempered
semitone.Both are derived from the
ratio of two frequencies, regardless of where they lie within the keyboard
compass.It is important to bear in
mind that the ratio of two frequencies is a dimensionless number with no
units. Unlike a beat, it does not express the difference between the
frequencies, which would be measured in units of Hz for example.

13.
In the course of some extremely helpful correspondence, M Moreau also mentioned
that the firm of Kleuker Orgelbau may have tuned organs to Cordier's
temperament. This temperament attracted favourable reviews from Jean
Guillou who wrote the Foreword to Cordier's book [11].